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Clarify some points about the project aim.

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Frederik Hanghøj Iversen 2018-01-15 17:56:08 +01:00
parent 69adb726de
commit 7090c2c6bf
2 changed files with 38 additions and 3 deletions
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5
proposal/macros.tex
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@ -4,6 +4,7 @@
\newcommand{\defeq}{\coloneqq} \newcommand{\defeq}{\coloneqq}
\newcommand{\bN}{\mathbb{N}} \newcommand{\bN}{\mathbb{N}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\to}{\rightarrow}} \newcommand{\to}{\rightarrow}}
\newcommand{\mto}{\mapsto}} \newcommand{\mto}{\mapsto}}
\newcommand{\UU}{\ensuremath{\mathcal{U}}\xspace} \newcommand{\UU}{\ensuremath{\mathcal{U}}\xspace}
@ -12,3 +13,7 @@
\newcommand{\todo}[1]{\textit{#1}} \newcommand{\todo}[1]{\textit{#1}}
\newcommand{\comp}{\circ} \newcommand{\comp}{\circ}
\newcommand{\x}{\times} \newcommand{\x}{\times}
\newcommand{\Hom}{\mathit{Hom}}
\newcommand{\fmap}{\mathit{fmap}}
\newcommand{\idFun}{\mathit{id}}
\newcommand{\Sets}{\mathit{Sets}}

36
proposal/proposal.tex
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@ -99,6 +99,26 @@ points. This is exactly the notion of equality of functions that we are
interested in; that they are equal for all inputs. We call this interested in; that they are equal for all inputs. We call this
\nomen{pointwise equality}, where the \emph{points} of a function refers \nomen{pointwise equality}, where the \emph{points} of a function refers
to it's arguments. to it's arguments.
In the context of category theory the principle of functional extensionality is
for instance useful in the context of showing that representable functors are
indeed functors. The representable functor for a category $\bC$ and a fixed
object in $A \in \bC$ is defined to be:
%
\begin{align*}
\fmap \defeq X \mapsto \Hom_{\bC}(A, X)
\end{align*}
%
The proof obligation that this satisfies the identity law of functors
($\fmap\ \idFun \equiv \idFun$) becomes:
%
\begin{align*}
\Hom(A, \idFun_{\bX}) = (g \mapsto \idFun \comp g) \equiv \idFun_{\Sets}
\end{align*}
%
One needs functional extensionality to ``go under'' the function arrow and apply
the (left) identity law of the underlying category to proove $\idFun \comp g
\equiv g$ and thus closing the above proof.
% %
\iffalse \iffalse
I also want to talk about: I also want to talk about:
@ -169,6 +189,15 @@ project will study and formalize this model. Note that I will \emph{not} aim to
formalize CTT itself and therefore also not give the formal translation between formalize CTT itself and therefore also not give the formal translation between
the type theory and the meta-theory. Instead the translation will be accounted the type theory and the meta-theory. Instead the translation will be accounted
for informally. for informally.
The project will formalize CwF's. It will also define what pieces of data are
needed for a model of CTT (without explicitly showing that it does in fact model
CTT). It will then show that a CwF gives rise to such a model. Furthermore I
will show that cubical sets are presheaf categories and that any presheaf
category is itself a CwF. This is the precise way by which the project aims to
provide a model of CTT. Note that this formalization specifcally does not
mention the language of CTT itself. Only be referencing this previous work do we
arrive at a model of CTT.
% %
\section{Context} \section{Context}
% %
@ -230,7 +259,7 @@ assistant.
One particular challenge in this context is that in a cubical setting there can One particular challenge in this context is that in a cubical setting there can
be multiple distinct terms that inhabit a given equality proof.\footnote{This is be multiple distinct terms that inhabit a given equality proof.\footnote{This is
in contrast with ITT that enjoys \nomen{Uniqueness of identity proofs} in contrast with ITT where one \emph{can} have \nomen{Uniqueness of identity proofs}
(\cite[p. 4]{huber-2016}).} This means that the choice for a given equality (\cite[p. 4]{huber-2016}).} This means that the choice for a given equality
proof can influence later proofs that refer back to said proof. This is new and proof can influence later proofs that refer back to said proof. This is new and
relatively unexplored territory. relatively unexplored territory.
@ -240,8 +269,9 @@ basics of. So learning the necessary concepts from Category Theory will also be
a goal and a challenge in itself. a goal and a challenge in itself.
After this has been implemented it would also be possible to formalize Cubical After this has been implemented it would also be possible to formalize Cubical
Type Theory and formally show that Cubical Sets are a model of this. This is not Type Theory and formally show that Cubical Sets are a model of this. I do not
a goal for this thesis but rather a natural extension of it. intend to formally implement the language of dependent type theory in this
project.
The thesis shall conclude with a discussion about the benefits of Cubical Agda. The thesis shall conclude with a discussion about the benefits of Cubical Agda.
% %